A Borsuk-Ulam theorem for adaptive continuous measurements

Abstract

We prove a Borsuk-Ulam theorem for adaptive measurements: if $2^n\le m$, then any $n$ adaptively chosen continuous functionals take identical values at two antipodal points of any sphere in $\mathbb{R}^m$. Consequently, at least $\lceil\log_2(m+1)\rceil$ adaptive continuous measurements are needed to learn the vectors of a ball in $\mathbb{R}^m$ with an error smaller than its radius, which matches the recently obtained upper bound up to one measurement. The bound extends to Bernstein widths in Banach spaces. The main idea of the proof was found by Claude Fable 5, an AI model developed by Anthropic, and we document the development in a last section.